tag:blogger.com,1999:blog-382677439644973005.post7711248633957579837..comments2024-08-19T10:24:07.345+02:00Comments on Physics intuitions: Exhaustion of nested squares and Wallis productArjen Dijksmanhttp://www.blogger.com/profile/09450431291713605237noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-382677439644973005.post-79324872919379141012019-01-27T17:37:23.670+01:002019-01-27T17:37:23.670+01:00I am happy you could find this page. Please share ...I am happy you could find this page. Please share also your thoughts about the Wallis productArjen Dijksmanhttps://www.blogger.com/profile/09450431291713605237noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-71064248852392260012018-08-24T00:45:14.406+02:002018-08-24T00:45:14.406+02:00Hi,
I have been thinking about a geometrical deriv...Hi,<br />I have been thinking about a geometrical derivation for the wallis product for some weeks. Today I came across this page and I am very happy about this solution!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-65352679496696178632010-03-22T20:42:10.941+01:002010-03-22T20:42:10.941+01:00Hi Phil,
That downward perspective rendering of a...Hi Phil,<br /><br />That downward perspective rendering of a pyramid is indeed interesting. Friends around me also pointed me towards that, when I showed them that figure. Concerning fractals, it seems indeed that nature is full of recurrent patterns from which we gain insight. I'll have to look closer at works from Von Koch ans Mandelbrot.<br /><br />Besides, as a complement to my post, I realized today that there is another geometrical representation for Wallis product. In fact I read it a week ago at a <a href="http://pballew.blogspot.com/2010/03/but-why.html" rel="nofollow">very entertaining blog</a>, but I didn't realize at that time that it was Wallis' product, because I was only beginning to learn about it.Arjen Dijksmanhttps://www.blogger.com/profile/09450431291713605237noreply@blogger.comtag:blogger.com,1999:blog-382677439644973005.post-72165105018611807322010-03-21T22:34:14.037+01:002010-03-21T22:34:14.037+01:00Hi Arjen,
I can see the wheels are turning or sho...Hi Arjen,<br /><br />I can see the wheels are turning or should I say the circles :-) Levity aside some of best ideas regarding mathematics, which do or may relate to our physical world, came as a result of such thinking, as to imagine geometrically how things come to be as they are. The circle of course probably being the best example of them all, combined with its three dimensional offspring being the sphere. <br /><br />What I find interesting in your first demonstrated analysis, that being besides what you mentioned, is what you end up with is a perspective rendering of a pyramid viewed from above. So it has us to wonder as we iterate in if we are also not interating up or perhaps down, as to have another dimension or rather partial (fractal) dimension come to be realized. <br /><br />As it happens others, first beginning with those such as <a href="http://www.gap-system.org/~history/Biographies/Koch.html" rel="nofollow">Von Koch</a>, with his snowflake would have the concept of <a href="http://www.jimloy.com/fractals/koch.htm" rel="nofollow">fractal dimensioning</a>, as you give light to here, have us come to conclusions about dimensions not being of increments of only whole numbers yet also as fractions of the whole. Of course <a href="http://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrot" rel="nofollow">Benoit Mandelbrot</a> would later discover this could be extended to the <a href="http://www.youtube.com/watch?v=J5ffjRI5VlU" rel="nofollow">the complex number plane</a> , where mathematical concepts like imaginary numbers come to be better understood as not rightfully referred to being imaginary. This has then had it easier for me to imagine where all those digits of PI could possible reside, if they only being restricted to existing in one dimension as it is the norm to have it so conceptualized.<br /><br />So then I’ve often wondered if Archmedes could ever imagined how his first thought born of drawing circles in the sand would lead in terms of our ever broadening undertstanding of mathematics yet the world itself more generally.<br /><br />Best,<br /><br />PhilPhil Warnellhttp://decartes-einstein.blogspot.com/noreply@blogger.com